Question

Write the standard form of the equation and the general form of the equation of each circle of radius \(r\) and center \((h, k)\). Graph each circle. $$ r=3 ;(h, k)=(0,0) $$

Short answer

Standard form: \(x^2 + y^2 = 9\). General form: \(x^2 + y^2 - 9 = 0\). Center at origin, radius 3.

Step-by-step solution

Identify the parameters

Given the radius of the circle as \( r = 3 \) and the center of the circle as \((h, k) = (0,0)\).

Write the standard form of the equation

The standard form of the equation of a circle with radius \(r\) and center \((h, k)\) is \((x-h)^2 + (y-k)^2 = r^2\). Substituting the given values, we get \((x-0)^2 + (y-0)^2 = 3^2\). Simplify to get the standard form: \(x^2 + y^2 = 9\).

Write the general form of the equation

The general form of the equation of a circle is \(Ax^2 + Ay^2 + Bx + Cy + D = 0\). For our circle, we expand and see that the general form is the same as the standard form but rearranged. Therefore, the general form is: \(x^2 + y^2 - 9 = 0\).

Graph the circle

To graph the circle, draw a coordinate plane. Since the center is at \((0, 0)\), place the center at the origin. Draw a circle with a radius of 3 units around this point.

Key concepts

standard form

The standard form of the equation of a circle is essential for understanding how a circle is drawn on a coordinate plane. It looks like this: \((x-h)^2 + (y-k)^2 = r^2\). To break it down:

\(h\): This is the x-coordinate of the circle's center.
\(k\): This is the y-coordinate of the circle's center.
\(r\): This is the radius of the circle.

So, when we plug in the given values — a radius \( r = 3 \) and a center \( (h,k)=(0,0) \) — our equation becomes: \((x-0)^2 + (y-0)^2 = 3^2\). This simplifies easily to \( x^2 + y^2 = 9 \).
This form is useful because it directly shows you the radius and the center of the circle, making it straightforward to graph.

general form

The general form of the equation of a circle can sometimes look more complicated, but it's just another way to represent the same information. The general form is written as: \(Ax^2 + Ay^2 + Bx + Cy + D = 0\).

Here:
\(A\): Coefficient for \(x^2\) and \(y^2\) — note that they have the same coefficient.
\(B\): Coefficient for \(x\) — often representing the horizontal shift.
\(C\): Coefficient for \(y\) — often representing the vertical shift.
\(D\): The constant term.

For the circle in our exercise with center \( (0,0) \) and radius \( r = 3 \), the standard form is \( x^2 + y^2 = 9 \). When we expand this equation to the general form, it remains \( x^2 + y^2 - 9 = 0 \). Notice that \( B \) and \( C \) are zero because there are no linear \( x \) or \( y \) terms. The know-how to flip between the standard and general forms enhances your understanding and flexibility in tackling different math problems.

radius and center

Understanding the radius and center of a circle is crucial for mastering circle equations. Let's break these down:

\( r \): This is the radius of the circle. It's the distance from the circle's center to any point on its circumference.
\(h\) and \(k\): These represent the coordinates of the circle's center on the coordinate plane — \( h \) being the x-coordinate, and \( k \) being the y-coordinate.

In our exercise, we have:
Replacing \( r \) with 3 shows that every point on the circle is exactly 3 units away from the origin. So, the center \( (h,k) = (0,0) \) is right on the origin, making it very straightforward to draw the circle. This simple understanding makes graphing the circle as easy as following the steps below:

• Draw a coordinate plane with x and y axes.
• Place the circle's center at (0,0).
• Draw a circle reaching 3 units away from the center in all directions.

This visual representation ties together how the equation reflects the actual geometric shape.